1

A NEW APPROACH TO GEAR VIBRATION DEMODULATION AND ITS APPLICATION TO DEFECT DETECTION

Jun Ma

Department of Mechanical Engineering New York Institute of Technology

Old Westbury, NY 11568

C. James Li Department of Mechanical Engineering

Rensselaer Polytechnic Institute Troy, NY 12180

Abstract: A new scheme for gear vibration demodulation is described. First, a gear vibration model that includes amplitude and phase modulations is proposed. Based on this model, an algorithm which considers all tooth meshing harmonics as modulation carriers is established. This algorithm derives amplitude and phase modulation signals from a gear's vibration signal average obtained by synchronized averaging. This iterative algorithm first finds an approximation to the phase modulation signal from the signal average. Using the phase modulation signal, the amplitude modulation signal is then identified from the signal average. The effectiveness of this scheme is confirmed and compared to the state-of-the-art using a vibration average of a gear which has two defects of different sizes. Key Words: Condition monitoring; Demodulation of gear vibrations; Gear defect detection; Gear vibration model; Signal analysis; Signal processing; Vibration analysis.

1. INTRODUCTION

For years, monitoring the condition of power transmissions has been deemed imperative. As indicated in an Army report (US Army, 1976), power plants and drive systems contributed to 68% of flight safety incidents related to failures in mechanical systems and to 58% of direct maintenance costs. Thus, gearboxes, the core of power transmission, have received considerable attention in the field of condition monitoring and fault diagnosis. In particular, gear localized defects have been extensively studied, since a large percent (60%) of gearbox damages are due to gear faults, which in turn are mostly initiated by localized defects. For example, a study on gear faults (Allianz, 1978) reported that three types of gear localized defects, namely, forced fracture, fatigue fracture, and incipient cracks, were responsible for about 90% of gear faults found in gearboxes (56% 17% , and 16%, respectively).

Vibrations externally measured on a gearbox have been used to monitor the operating condition of the gearbox and diagnose the fault, if there is any, without interfering with the normal operation. The most common method employed for examining mechanical vibration is spectral analysis, by which defects such as eccentricity or local tooth damage are expected to be identified by increases of modulation sidebands

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in the spectrum. These sidebands are located on both sides of gear tooth meshing frequency and its harmonics, and are separated by integer multiples of gear rotation frequency. For example, the ratio of the sideband power to the carrier (tooth meshing frequency) power, called SBratio, was investigated (Dousis, 1986). However, the recognition of modulation sidebands is difficult due to the large number of gears, rotating at different speeds, contained in a gearbox. Moreover, the sensitivity of spectral sidebands to localized defects is low because modulations produced by a localized defect are transient events (Randall, 1982) and consequently, are inherently unsuitable for spectral analysis which assumes stationarity of signals.

In addition to spectral analysis, some statistical parameters, such as non-dimensionalized sixth moment (Astridge, 1986) and some other mean value based indices (Rose, 1990), have been established to assess the condition of gears from their vibrations. This type of technique usually has to be applied together with a trend analysis and thus the whole history of parameters must be monitored. Unfortunately, as the modulation of short duration does not change statistical properties of the overall vibration by much, they are frequently insensitive to localized defects, especially to incipient ones. Furthermore, most of these statistical parameters are dependent on the operation conditions such as load and speed, and consequently have limited practical uses.

To a large extent, vibrations produced solely by a gear and its carrying shaft exhibit a repetitive pattern from one rotation to the next. Thus, there is a strong correspondence between time domain features of the vibration signal and angular positions of the rotating shaft. To obtain this angular position dependent signal from noisy measurements, the synchronized signal averaging (Braun and Seth, 1979), which has the effect of a comb filter, maybe employed. Vibrations of the gear of interest are sampled at the same angular positions for a number of rotations and samples corresponding to the same angular position are then averaged over the rotations. This widely adopted pre-processing procedure offers noise reduction and removal of interferences from components rotating at different speeds. The result, called signal average, is a function of angular position that shows the repetitive vibration pattern of tooth meshing, including any modulation, over one rotation.

For advanced gear defects, a simple visual inspection of the signal average maybe sufficient to detect the damage, as in a case that will be shown shortly (Fig. 2). But the detection of defects at a very early stage requires sophisticated signal detection and processing techniques to enhance the defect information contained in the signal average. Due to the modulation nature of defect related vibration, a number of existing gear defect detection techniques consider amplitude and phase modulations as defect signature. It has been found (McFadden and Smith, 1985) that by bandpass filtering the signal average about the largest meshing harmonic, eliminating that harmonic and then enveloping, a signal considered representing the modulation energy could be obtained. A further study (McFadden, 1986) tried to demodulate the signal average around the largest meshing harmonic to provide both the approximate amplitude and phase signals. However, in all these studies, only a single tooth meshing harmonic and its sidebands were utilized to find the modulation signals,

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while, in fact, all meshing harmonics carry information about the modulation signal. Consequently, a lot of useful modulation information were ignored or misinterpreted.

In this paper a new gear vibration demodulation scheme is described. As opposed to the narrowband approach mentioned above, the scheme takes all tooth meshing harmonics into consideration as modulation carriers and recovers amplitude and phase modulation signals from the wide band signal. Since gear localized defects produce local modulations to gear vibrations, they can be easily identified from modulation signals by inspection. The effectiveness of the scheme in gear localized defect detection is examined with the vibration average of a gear containing both an advanced defect and a smaller defect.

2. GEAR VIBRATION MODEL: MODULATIONS

This section presents a gear vibration model based on which a demodulation scheme will be derived. To establish such a model, let us consider a pair of perfect mating gears whose teeth are rigid with exact involute profile, and are equally spaced. Such a pair of gears would transmit exactly uniform angular motion in the absence of distributed defects such as runout, imbalance, and misalignment. Any deviation from this ideal situation will cause variations in both angular displacement and velocity, i.e., the transmission error, which in turn results in variations in the force transmitted between the meshing teeth. Vibrations will then be generated and transmitted everywhere in and on the gearbox through the gear-bearing-shaft-bearing-casing path.

Then consider a pair of gears whose teeth are not rigid but otherwise the same as the aforementioned perfect gears, meshing under constant load at constant speed. Since the contact stiffness varies periodically with the number of teeth in contact and with the contacting position on tooth surface, vibration will be excited at tooth meshing frequency. After being synchronously averaged, the vibration of this pair of gears may be approximately represented in terms of tooth meshing frequency fm and its harmonics:

xt (t ) = Xk cos(2πkf mt + φk)

k=0

K

∑ (1)

As mentioned in the previous section, the vibration can be viewed as a function of angular position θ. Therefore, we shall express the vibration of a gear with N teeth as:

x(θ) = Xk cos(kNθ + φk )

k=0

K

∑ (2)

Now take into consideration the tooth profile error, tooth spacing error, and defects. All these will vary contact stiffness and therefore produce changes in the amplitude and phase of the vibration at meshing frequency and its harmonics as they go through

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the meshing. Let these changes be described by the amplitude and phase modulating functions ak(θ) and bk(θ), then the modulated vibration is

y(θ ) = 1 + ak (θ)[ ]Xk cos kNθ + φk + bk(θ)[ ]

k=0

K

∑ (3)

The amplitude modulating function changes the envelope of x(θ), and therefore should be independent of tooth meshing harmonics. On the other hand, since the phase modulating signal at any given instant produces the same time delay to all tooth meshing harmonics in the vibration, it is linearly proportional to the harmonic number k. Thus Eq. (3) is simplified into:

y(θ ) = 1 + a(θ )[ ] Xk cos kNθ + φk + kb(θ)[ ]

k= 0

K

∑ (4)

where a(θ) and b(θ) themselves can be expanded into Fourier series in harmonics of rotational frequency of the gear:

a(θ) = Ap cos(pθ + α p )

p=1

P

∑ (5)

b(θ ) = Bq cos(qθ +β q )

q=1

Q

∑ (6)

This is because, obviously, they repeat themselves from one rotation to the next, as the signal average does.

If the modulating functions a(θ) and b(θ) can be extracted from the signal average y(θ), then we will be able to evaluate the gear condition and detect any defects on the gear because information on tooth variations from average is carried in a(θ) and b(θ). In the following section, an algorithm for extracting modulation signals from the signal average is derived and applied to gear localized defect detection.

3. DEMODULATION METHOD

3.1 Separation of Phase and Amplitude Modulations: Suppose that amplitude and phase modulating signals a(θ) and b(θ) are narrow-band signals, containing no components with frequency higher than the tooth meshing frequency, i.e., 0<P,Q<N. Then the analytic signal ya(θ) of real measurement y(θ) can be approximated by (Rihaczek, 1966)

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ya (θ) ≈ 1 + a(θ )[ ] Xk exp j ⋅ kNθ + φk + kb(θ )[ ]{ }

k=0

K

∑ (7)

where j is the complex numeric, and ya(θ) has y(θ) as its real part and the Hilbert transform ˆ y (θ ) of y(θ) as its imaginary part:

ya (θ) = y(θ ) + j ⋅ ˆ y (θ ) (8)

ˆ y (θ ) =

1π

y(ϑ )θ − ϑ

dϑ∫ (9)

Let

z(θ) = Xk exp j ⋅ kNθ + φk + kb(θ )[ ]{ }

k=0

K

∑ (10)

then Eq. (7) becomes:

ya (θ) ≈ 1 + a(θ )[ ]⋅ z(θ ) (11)

Note that the real signal [1+a(θ)] does not contribute phase angle to the right hand side of Eq. (7). Therefore, complex signals ya(θ) and z(θ) have the same phase angle for all θ. If we know Xk and φk, we can find b(θ) by equating phase angles of ya(θ) and z(θ), and then a(θ) can be solved from Eq. (11) by

a(θ) =

ya(θ )z(θ )

−1 (12)

However, Xk and φk are not readily available. In the following, an analysis is given to find the relationship between these quantities and modulation signals. Then a demodulation scheme will be derived.

3.2 Meshing Component Retrieving: Instead of having Xk and φk , what we have are Fourier magnitudes Yl and phase angles ϕl of the signal average y(θ) through its Fourier expansion

y(θ ) = Yl cos(lθ + ϕl )

l =0

L

∑ (13)

To obtain Xk and φk from y(θ), we compare tooth meshing harmonic in the model (4) and the measurement described in Eq. (13), which are two different expressions of the same signal.

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Assume that the influence on a carrier, i.e., one of the tooth meshing harmonics, from modulation sidebands of its two neighboring carrier harmonics is negligible. This means that sidebands of a tooth meshing harmonic caused by modulations are negligible when they are away from that tooth meshing harmonic by more than the tooth meshing frequency. Then the modulation effect on each carrier component may be examined individually. In other words, for a particular pair of Xk and φk, the modulation on the k-th meshing harmonic, which is the k-th term in model (4)

1+ a(θ)[ ]Xk cos kNθ + φk + kb(θ )[ ] will be examined. Noting that amplitude modulation does not change the magnitude and phase of the carrier, i.e., the meshing harmonic, we can consider only the phase modulation for examining the carrier component. Without amplitude modulation, the k-th term in Eq. (4) becomes the following:

v(θ) = Xk cos kNθ + φk + kb(θ)[ ]

= Vi cos(iθ + γ i )i =0

I

∑ (14)

When i=kN, Vi and γi give the Fourier magnitude and phase at the k-th meshing harmonic, which should be equal to those given by the measurement y(θ):

VkN = YkN (15)

γ kN = ϕkN (16)

On the other hand, Xk and φk.can be calculated from VkN and γkN by the following two equations (see Appendix):

Xk =

VkNMk (17)

φk = γkN − pk (18)

However Mk and pk are related with Fourier magnitudes and phases of b(θ) through

Mk ⋅ exp( jpk ) = Jnq

(c ) (kBq )q=1

Q

∏⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ c

∑ exp j nq(c )(βq +

π2

)q=1

Q

∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

(19)

where n1(c)

,n2(c)

,..., nQ(c)

are permutations of n1, n2, ..., nQ satisfying

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nq

(c)q = 0q =1

Q

∑ (20)

and Jn(x) is the Bessel function of the first type.

Therefore, if the phase modulating signal b(θ) is known, i.e., Bq and βq are known, Mk and pk can be obtained by solving Eq. (19), and then Xk and φk can be recovered using Eqs. (17) and (18).

3.3 Proposed Scheme: Above analysis provides a means to derive Xk and φk from the signal average when b(θ) is given. These Xk and φk can be used, in turn, to obtain another b(θ) by equating phase angles of two sides of Eq. (11), i.e., comparing the phase of ya(θ) and z(θ). If the initially given b(θ) is the actual one, it will be the same as the newly obtained one. Using these two relationships iteratively, a procedure can be established to find the true b(θ). In the following, this iterative procedure is detailed (see Fig. 1):

(a) Calculate the Fourier magnitudes Yl and phases ϕl of signal average y(θ) at tooth meshing harmonics, i.e., l=kN;

(b) Using Hilbert transform, construct the analytic signal ya(θ) from y(θ) as described in Eqs. (8) and (9);

(c) Make an initial guess on b(θ); (d) Find Mk and pk for all k's according to Eq. (19); (e) Calculate Xk and φk using Eqs. (17) and (18), then construct the

complex z(θ) by Eq. (10); (f) Compare the phase of ya(θ) and z(θ). If the error is small enough,

continue; otherwise, adjust b(θ) and go back to Step (d); (g) Calculate a(θ)using Eq. (12).

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Initial b(θ) — Eq. (22)

New b(θ) — Powell's Method

z(θ) — Eq. (10)

a(θ) — Eq. (12)

arg z(θ) = arg y (θ)a

Yes

No

M , p — Eq. (19)kk

X , φ — Eqs. (17), (18)k k

Fig. 1 Diagram of the full-spectrum demodulation procedure

In our implementation of the scheme, Powell's method (e.g. Wismer and Chattergy, 1978) is employed in Step (f) to search for the b(θ) that minimizes

J b(θ)( ) = arg ya(θ)( )− arg z(θ )( )[ ]2 dθ

0

2π∫ (21)

To reduce the computational burden, discrete Fourier components of b(θ), instead of b(θ) itself, are searched to minimize J(b(θ)). Depending on the initial guess of b(θ), the search algorithm may converge to different local minimum of the objective function (21) and the convergence time is different, too. The bandpass-demodulation method that considers the largest meshing harmonic only (McFadden, 1986) can be used to obtain a good guess for presumably, the largest carrier harmonic carries the largest fraction of modulation energy. Specifically, the signal average is bandpass filtered about the largest tooth meshing harmonic, say the k-th. Then an analytic signal ya(bp) is constructed for the bandpassed signal average through Hilbert transform. The initial guess b0(θ) can then be obtained by

b0(θ) =

1k

arg ya(bp)[ ]− Nθ (22)

4. EXPERIMENTAL INVESTIGATION

The experimental setup, as illustrated in Fig. 2, contains a three-stage reduction gearbox driven by a variable speed DC motor. The load is provided by a disk brake and is adjustable with a screw-spring loading mechanism. The 20-tooth input pinion has two artificially seeded defects of different size at teeth 11 and 18, respectively, simulating fractured teeth at different stage. Since modulation sidebands can be

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easily distorted by the effect of vibration transmission path (McFadden and Smith, 1986), torsional vibration, instead of translational vibration, is measured using a Hoodwin 34E angular acceleration sensor based on the principle that the eddy current induced in a disk armature is proportional to the rotational speed of the disk (Hoodwin, 1967). To facilitate the synchronous averaging, a 450-line optical encoder is employed to trigger the sampling of the vibration signal.

Variable speed DC motor

3-stage spur reduction gearbox

Brake assembly

and load cell

Optical encoder

Torsional accelerometer

assembly

Signal filtering, sampling, and processing

Fig. 2 Experimental setup for gear defect detection

Fig. 3 shows a signal average, obtained from 200 rotations, as a function of angular position (tooth number), and its spectrum indexed by the rotational frequency (order). It is obvious that an advanced fracture, like the one on tooth 11, can be detected immediately by a visual inspection of the signal average. The larger amplitude of vibration around that tooth indicates reduced meshing stiffness. However, the smaller defect on tooth 18 has little noticeable effect on the signal average. Additionally, the band-pass demodulation method (McFadden, 1986), which will be used later to provide an initial guess about phase modulation signal, also failed to detect the defect (Fig. 4).

10

-200

0

200

0 2 4 6 8 10 12 14 16 18 20

Vibration Average

tooth number

0

5

10 x107

0 10 20 30 40 50 60 70 80 90 100

Power Spectrum

order

Fig. 3 Vibration signal average and its spectrum

-2

0

2

4

0 2 4 6 8 10 12 14 16 18 20

"amplitude modulation signal"

tooth number

-5

0

5

0 2 4 6 8 10 12 14 16 18 20

"phase modulation signal"

tooth number

Fig. 4 Results by bandpass-demodulation method

To evaluate the proposed scheme, we applied it to the signal average. The proposed method assumes that a phase modulation signal can be approximated by a Fourier series that contains up to Q=N-1 (here N=20) rotational harmonics. The larger the Q, the better the approximation is to the phase modulation, and the more computation is required. For example, the amount of computing power required for solving Eq. (20) for permutations nq(c), q=1,2,...,Q, would grow exponentially with respect to Q. To find how sensitive the scheme is to different Q's, we implemented the scheme with Q=7 and Q=5, respectively. It was found that both phase and amplitude modulation

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signals were very similar in both cases. Since only a small number (5 or 7, compared to 19 which was assumed maximum number in this case) of rotational harmonics in phase modulation signal were considered, the amplitude modulation signal should give better indication to gear condition than phase modulation signal could. In fact, defects on tooth 11 and tooth 18 were obvious in the amplitude modulation signal solved with Q=7 which is much lower than the N-1(see Fig. 5).

-200

0

200

0 2 4 6 8 10 12 14 16 18 20

(a) signal average

tooth number

-2

0

2

0 2 4 6 8 10 12 14 16 18 20

(b) "amplitude modulation signal"

tooth number

Fig. 5 Vibration signal average and amplitude modulation signal

5. CONCLUSIONS

Torsional vibrations produced by a gear containing localized defects were examined. A gear vibration model which includes angular and amplitude modulations was proposed. Means for constructing this model and then recovering the angular and amplitude modulation signals from real signal average was established. It has been found that the amplitude modulation signal recovered by the scheme is very sensitive to defects compared to the bandpass demodulation scheme, although phase modulation signal does provide some indications to defects as well. Why the phase modulation is less sensitive to the defects is due to the fact that we restrained ourselves from approximating it with large number of Fourier components to alleviate the demand on computing power. Nevertheless, the amplitude modulation signal is very effective in indicating local modulations that are associated with localized defects.

The superiority of the proposed full spectrum demodulation method over the state-of-the-art has been demonstrated by experimental study on a gear with two localized defects of different sizes. This method successfully detected the advanced defect as well as the smaller one, which is not detectable to the bandpass demodulation

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method. It can be used for gear quality evaluation in manufacturing and gear condition assessment and localized defect detection in on-line applications.

References

Allianz Versicherungs-AG, 1978, Handbook of Loss Prevention, Springer-Verlag, Berlin

Astridge, D. G., 1986, "Vibration Health Monitoring of the Westland 30 Helicopter Transmission — Development and Service Experience", Detection, Diagnosis and Prognosis of Rotating Machinery to Improve Reliability, Maintainability, and Readiness through the Application of New and Innovative Techniques, Proceedings of the 41st Meeting of the Mechanical Failures Prevention Group, T. R. Shives and L. J. Mertaugh ed., Cambridge University Press, New York, pp. 200-215

Braun, S., and Seth, 1979, "On the Extraction and Filtering of Signals Acquired from Rotating Machines", Journal of Sound and Vibration, Vol. 65, No. 1, pp. 37-50

Dousis, D., A., 1986, "Gear Failure Analyses in Helicopter Main Transmissions Using Vibration Signature Analysis", Detection, Diagnosis and Prognosis of Rotating Machinery to Improve Reliability, Maintainability, and Readiness through the Application of New and Innovative Techniques, Proceedings of the 41st Meeting of the Mechanical Failures Prevention Group, T. R. Shives and L. J. Mertaugh ed., Cambridge University Press, New York, pp. 133-144

Hoodwin, L. S., 1967, "Angular Accelerometer: Advantages and Limitations", Instruments and Control Systems, April 1967, p. 129

McFadden, P. D., 1986, "Detecting Fatigue Cracks in Gears by Amplitude and Phase Demodulation of the Meshing Vibration", Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 108, No. 2, pp. 165-170

McFadden, P. D. and Smith, J. D., 1985, "A Signal Processing Technique for Detecting Local Defects in Gear from the Signal Average of the Vibration", Proceedings of Institute of Mechanical Engineers, Vol. 199, No. C4, pp. 287-292

McFadden, P. D., and Smith, J. D., 1986, "Effect of Transmission Path on Measured Gear Vibration", Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 108, pp. 377-378

Randall, R. B., 1982, "A New Method of Modeling Gear Faults", Journal of Mechanical Design, Vol. 104, pp. 259-267

Rihaczek, A. W., 1966, "Hilbert Transforms and the Complex Representation of Real Signals", Proceedings of IEEE, Vol. 54, pp. 434-435

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Rose, H. J., 1990, "Vibration Signature and Fatigue Crack Growth Analysis of a Gear Tooth Bending Fatigue Failure", Current Practices and Trends in Mechanical Failure Prevention, Proceedings of the 44th Meeting of the Mechanical Failures Prevention Group, H. C. Pusey and S. C. Pusey ed., Vibration Institute, Willowbrook, NJ, pp. 235-245

US Army, 1976, "Update to Reliability and Maintainability Planning Guide for Army Aviation Systems and Components", USAAVSCOM Technical Report 77-15

Wismer, D. A., and Chattergy, R., 1978, Introduction to Nonlinear Optimization, A Problem Solving Approach, North-Holland, New York

Appendix CARRIER COMPONENTS IN PHASE MODULATION

When a sinusoidal function of angular position θ, of frequency kN with magnitude Xk and phase φk, is phase-modulated by another function kb(θ), the resulted signal can be alternatively expressed in many ways:

v(θ) = Xk cos kNθ + φk + kb(θ)[ ]= Xk Re exp j kNθ + φk + kb(θ)( )[ ]{ }

= Xk Re exp j kNθ + φk + k Bq cos(qθ +β q )q=1

Q

∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

= Xk Re exp j kNθ + φk + k Bq sin(qθ + βq +π2

)q=1

Q

∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

= Xk Re exp j kNθ + φk( )[ ]⋅ exp j kBq sin(qθ +β q + π2

)⎛ ⎝

⎞ ⎠

⎡

⎣ ⎢ ⎤

⎦ ⎥ q=1

Q

∏⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ (A1) where j is the complex numeric, Bq and βq are Fourier magnitudes and phases of b(θ), respectively. In order to investigate the modulation effect of b(θ) on the carrier component, let us first consider the modulation effect of only one Fourier component of b(θ), say the q-th component Bq cos(qθ +βq ) The modulated signal is

vq (θ ) = Xk Re exp j kNθ + φk( )[ ]⋅ exp j kBq sin(qθ +βq +

π2

)⎛ ⎝

⎞ ⎠

⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭ (A2)

The second exponential function on the right hand side is periodical with period 2π/q, so it has the following Fourier series representation:

exp j kBq sin(qθ +β q +

π2

)⎛ ⎝

⎞ ⎠

⎡ ⎣ ⎢

⎤ ⎦ ⎥ = Θn exp( jnqθ )

n=−∞

∞

∑ (A3)

14

where

Θn =1

2πexp j kBq sin(qθ +β q +

π2

)⎛ ⎝

⎞ ⎠

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ⋅ exp(− jnqθ )d(qθ )

−π

π∫

= exp jn βq +π2

⎛ ⎝

⎞ ⎠

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ⋅

12π

exp j kBq sin η( )[ ]⋅ exp(− jnη)dη−π

π∫

= exp jn βq + π2

⎛ ⎝

⎞ ⎠

⎡

⎣ ⎢ ⎤

⎦ ⎥ ⋅ Jn (kBq ) (A4)

where Jn(x) is the Bessel function of the first type, which has the following properties: (1) Jn(x) is a real valued function; (2) J n(x) = J−n (x) for even integer n and J n(x) = − J −n (x) for odd integer n;

(3) Jn

2 (x)n=−∞

∞

∑ = 1.

Substitute (A3), (A4) into (A2),we have:

vq (θ ) = Xk Re exp j kNθ + φk( )[ ]⋅ Jn (kBq )exp jn qθ +β q +π2

⎛ ⎝

⎞ ⎠

⎡

⎣ ⎢ ⎤

⎦ ⎥ n=−∞

∞

∑⎧ ⎨ ⎩

⎫ ⎬ ⎭

= Xk Re J n(kBq ) ⋅ exp j (kN + nq)θ + φk + n(β q +π2

)⎛ ⎝

⎞ ⎠

⎡

⎣ ⎢ ⎤

⎦ ⎥ n=−∞

∞

∑⎧ ⎨ ⎩

⎫ ⎬ ⎭ (A5)

Therefore, the carrier component at frequency kN (thus n=0) is modulated from Xk and φk to J0(kBq)Xk and φk. Similarly, for the most general case (A1), it can be obtained by induction that

v(θ) = Xk Re J n1(kB1)

n1

∑ Jn2(kB2 )

n2

∑ JnQ(kBQ )

nQ

∑⎧ ⎨ ⎪

⎩ ⎪

⋅exp j (kN + nqqq=1

Q

∑ )θ + φk + nq (βq + π2

)q=1

Q

∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎫ ⎬ ⎪

⎭ ⎪ (A6) This is saying that the carrier component has been affected jointly by all modulating harmonics. Specifically, if we express v(θ) by

v(θ) = Vi cos(iθ + γ i )

i=0

I

∑ (A7)

then VkN = Mk ⋅ Xk (A8) γ kN = pk + φk (A9) respectively, where Mk and pk satisfy

Mk ⋅ exp( jpk ) = Jnq

(c ) (kBq )q=1

Q

∏⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ c

∑ exp j nq(c )(βq +

π2

)q=1

Q

∑⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

(A10)

15

and n1(c)

,n2(c)

,..., nQ(c)

are permutations of n1, n2, ..., nQ satisfying

nq

(c)q = 0q =1

Q

∑. (A11)